Optimal. Leaf size=75 \[ \frac{16 a^2 \sqrt{a x^2+b x^3}}{15 b^3 x}-\frac{8 a \sqrt{a x^2+b x^3}}{15 b^2}+\frac{2 x \sqrt{a x^2+b x^3}}{5 b} \]
[Out]
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Rubi [A] time = 0.160626, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 a^2 \sqrt{a x^2+b x^3}}{15 b^3 x}-\frac{8 a \sqrt{a x^2+b x^3}}{15 b^2}+\frac{2 x \sqrt{a x^2+b x^3}}{5 b} \]
Antiderivative was successfully verified.
[In] Int[x^3/Sqrt[a*x^2 + b*x^3],x]
[Out]
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Rubi in Sympy [A] time = 19.0388, size = 66, normalized size = 0.88 \[ \frac{16 a^{2} \sqrt{a x^{2} + b x^{3}}}{15 b^{3} x} - \frac{8 a \sqrt{a x^{2} + b x^{3}}}{15 b^{2}} + \frac{2 x \sqrt{a x^{2} + b x^{3}}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**3+a*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0259506, size = 42, normalized size = 0.56 \[ \frac{2 \sqrt{x^2 (a+b x)} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3 x} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/Sqrt[a*x^2 + b*x^3],x]
[Out]
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Maple [A] time = 0.009, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 3\,{b}^{2}{x}^{2}-4\,abx+8\,{a}^{2} \right ) x}{15\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^3+a*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 1.39806, size = 57, normalized size = 0.76 \[ \frac{2 \,{\left (3 \, b^{3} x^{3} - a b^{2} x^{2} + 4 \, a^{2} b x + 8 \, a^{3}\right )}}{15 \, \sqrt{b x + a} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(b*x^3 + a*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218333, size = 54, normalized size = 0.72 \[ \frac{2 \,{\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt{b x^{3} + a x^{2}}}{15 \, b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(b*x^3 + a*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x^{2} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**3+a*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{b x^{3} + a x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(b*x^3 + a*x^2),x, algorithm="giac")
[Out]